arXiv subjects

Rickard Cullman

Publications and source records attributed to Rickard Cullman.

4 recordsLinked to original sources

arXiv2026

Primitive Averages, Directional Expansivity, and Quantitative Twisted Recurrence for Ergodic $\mathbb{Z}^d$-Actions

We prove two new results about probability preserving actions $T:\mathbb{Z}^d \curvearrowright (X,\mu)$. First, for a function $f\in L^2(\mu)$, we provide an explicit formula for the $L^2(\mu)$-limit of the average \[\frac{1}{|Q_N^\mathcal{P}|}\sum_{v \in Q_N^\mathcal{P}} T_v f\] where $\mathcal{P}\subset \mathbb{Z}^d$ is the set of primitive vectors, i.e. those for which the greatest common divisor of its components is $1$, and $Q_N^\mathcal{P}= [-N,N]^d\cap \mathcal{P}$. Second, for a set $A\subset X$ with $\mu(A)>0$, we provide a spectral condition under which the set of $\varepsilon$-expansive directions \[\left\{ v\in \mathbb{Z}^d \, : \, \mu\left(\bigcup_{n\in \mathbb{Z}} T_{nv}A\right)>1-\varepsilon\right\}\] has lower density very close to $1$. As an application of our techniques we are also able to prove a quantitative variant of a twisted multiple recurrence theorem of Bj\"orklund, Fish and the first author (arXiv:2503.02501).

math.DS↗

arXiv2025

Locally integrable cross sections and their intersection covolume

We study systematically cross sections of probability preserving actions of unimodular groups and their associated transverse measures, and introduce the invariant \emph{intersection covolume} to quantify their periodicity. Our main theorem, derived from a higher order version of Kac's lemma, shows that the intersection covolume is bounded below by the intensity, with equality precisely when the action is induced by a lattice (in the sense of Mackey). We further prove that the natural cross sections of cut--and--project actions have finite intersection covolume.

math.DS↗

arXiv2025

Periodicity of point processes in abelian groups without lattices

We investigate the intersection covolume of cross sections for probability preserving actions of a class of abelian groups without lattices, including $p$-adic groups and the group of finite adeles. We show that for cross sections with a uniformly discrete return time set, the intersection covolume is bounded below by twice the intensity, revealing a strict gap compared to the lattice case. Our main theorem asserts that cut-and-project systems uniquely attain the minimal intersection covolume. As an application, we characterize the generalized Farey fractions in the finite adeles via their Banach density.

math.DS↗

arXiv2025

Ehrhart spectra of large subsets of $\mathbb{Z}^r$

This paper introduces and studies the Ehrhart spectrum of a set $E \subseteq \mathbb{Z}^r$, defined as the set of all Ehrhart polynomials of simplices with vertices in $E$, generalizing the notion of volume spectrum. We show that for any $E \subseteq \mathbb{Z}^r$ with positive upper Banach density, there is some $n \in \mathbb{Z}^r$ such that the Ehrhart spectrum of $n \mathbb{Z}^r$ is contained in the Ehrhart spectrum of $E$, generalizing an earlier result by the first and third author for the volume spectrum of $E$.

math.DS↗